Analytical Treatment of the Free Vibration of Single-Walled Carbon Nanotubes Based on the Nonlocal Flugge Shell Theory

In the current work, the vibration characteristics of single-walled carbon nanotubes (SWCNTs) under different boundary conditions are investigated. A nonlocal elastic shell model is utilized, which accounts for the small scale effects and encompasses its classical continuum counterpart as a particular case. The variational form of the Flugge type equations is constructed to which the analytical Rayleigh–Ritz method is applied. Comprehensive results are attained for the resonant frequencies of vibrating SWCNTs. The significance of the small size effects on the resonant frequencies of SWCNTs is shown to be dependent on the geometric parameters of nanotubes. The effectiveness of the present analytical solution is assessed by the molecular dynamics simulations as a benchmark of good accuracy. It is found that, in contrast to the chirality, the boundary conditions have a significant effect on the appropriate values of nonlocal parameter.